1. Locus and Complex Numbers

2. Locus and Complex Numbers
  f  z  , find the locus of  or z
given some condition for  or z

3. Locus and Complex Numbers
  f  z  , find the locus of  or z
given some condition for  or z
(Make the condition the subject)

4. Locus and Complex Numbers
  f  z  , find the locus of  or z
given some condition for  or z
(Make the condition the subject)
 is purely real  Im   0, arg   0 or 

5. Locus and Complex Numbers
  f  z  , find the locus of  or z
given some condition for  or z
(Make the condition the subject)
 is purely real  Im   0, arg   0 or 
 is purely imaginary  Re   0, arg   

2

6. Locus and Complex Numbers
  f  z  , find the locus of  or z
given some condition for  or z
(Make the condition the subject)
 is purely real  Im   0, arg   0 or 
 is purely imaginary  Re   0, arg   

2
 linear function 
arg 
    locus is an arc of a circle
 linear function 

7. Locus and Complex Numbers
  f  z  , find the locus of  or z
given some condition for  or z
(Make the condition the subject)
 is purely real  Im   0, arg   0 or 
 is purely imaginary  Re   0, arg   

2
 linear function 
arg 
    locus is an arc of a circle
 linear function 

* minor arc if  
* major arc if  
2

* semicircle if  
2

2

8. z2
e.g .i  Find the locus of w if w 
,z 4
2

9. z2
e.g .i  Find the locus of w if w 
,z 4
2
z2
w
z
zw  z  2

10. z2
e.g .i  Find the locus of w if w 
,z 4
2
z2
w
z
zw  z  2
z w  1  2

11. z2
e.g .i  Find the locus of w if w 
,z 4
2
z2
w
z
zw  z  2
z w  1  2
2
z
w  1

12. z2
e.g .i  Find the locus of w if w 
,z 4
2
z2
w
z
zw  z  2
z w  1  2
2
z
w  1
2

4
w  1

13. z2
e.g .i  Find the locus of w if w 
,z 4
2
z2
w
z
zw  z  2
z w  1  2
2
z
w  1
2

4
w  1
2
4
w 1

14. z2
e.g .i  Find the locus of w if w 
,z 4
2
z2
w
z
zw  z  2
z w  1  2
2
z
w  1
2

4
w  1
2
4
w 1
w 1 
1
2

15. z2
e.g .i  Find the locus of w if w 
,z 4
2
z2
w
z
zw  z  2
z w  1  2
2
z
w  1
2

4
w  1
2
4
w 1
w 1 
1
2
1
 locus is a circle, centre 1,0  and radius
2
1
2
2
i.e.  x  1  y 
4

16. z 1
ii  Find the locus of z if w 
and w is purely real
z 1

17. z 1
ii  Find the locus of z if w 
and w is purely real
z 1
 x  1  iy  x  1  iy
w

 x  1  iy  x  1  iy

18. z 1
ii  Find the locus of z if w 
and w is purely real
z 1
 x  1  iy  x  1  iy
w

 x  1  iy  x  1  iy
x

2
 1  i x  1 y  i x  1 y  y 2
 x  12  y 2

19. z 1
ii  Find the locus of z if w 
and w is purely real
z 1
 x  1  iy  x  1  iy
w

 x  1  iy  x  1  iy
x

2
 1  i x  1 y  i x  1 y  y 2
 x  12  y 2
If w is purely real then Imw  0

20. z 1
ii  Find the locus of z if w 
and w is purely real
z 1
 x  1  iy  x  1  iy
w

 x  1  iy  x  1  iy
x

2
 1  i x  1 y  i x  1 y  y 2
 x  12  y 2
If w is purely real then Imw  0
i.e.   x  1 y   x  1 y  0

21. z 1
ii  Find the locus of z if w 
and w is purely real
z 1
 x  1  iy  x  1  iy
w

 x  1  iy  x  1  iy
x

2
 1  i x  1 y  i x  1 y  y 2
 x  12  y 2
If w is purely real then Imw  0
i.e.   x  1 y   x  1 y  0
 xy  y  xy  y  0

22. z 1
ii  Find the locus of z if w 
and w is purely real
z 1
 x  1  iy  x  1  iy
w

 x  1  iy  x  1  iy
x

2
 1  i x  1 y  i x  1 y  y 2
 x  12  y 2
If w is purely real then Imw  0
i.e.   x  1 y   x  1 y  0
 xy  y  xy  y  0
 2y  0
y0

23. z 1
ii  Find the locus of z if w 
and w is purely real
z 1
 x  1  iy  x  1  iy
w

 x  1  iy  x  1  iy
x

2
 1  i x  1 y  i x  1 y  y 2
 x  12  y 2
If w is purely real then Imw  0
i.e.   x  1 y   x  1 y  0
 xy  y  xy  y  0
 2y  0
y0
 locus is y  0, excluding 1,0 
 z  1  0, bottom of fraction  0 

24. z 1
ii  Find the locus of z if w 
and w is purely real
z 1
 x  1  iy  x  1  iy OR If w is purely real then arg w  0 or 
w

 x  1  iy  x  1  iy
x

2
 1  i x  1 y  i x  1 y  y 2
 x  12  y 2
If w is purely real then Imw  0
i.e.   x  1 y   x  1 y  0
 xy  y  xy  y  0
 2y  0
y0
 locus is y  0, excluding 1,0 
 z  1  0, bottom of fraction  0 

25. z 1
ii  Find the locus of z if w 
and w is purely real
z 1
 x  1  iy  x  1  iy OR If w is purely real then arg w  0 or 
w

 x  1  iy  x  1  iy
 z  1   0 or 
x 2  1  i x  1 y  i x  1 y  y 2
i.e. arg


2
 z 1
 x  1  y 2
If w is purely real then Imw  0
i.e.   x  1 y   x  1 y  0
 xy  y  xy  y  0
 2y  0
y0
 locus is y  0, excluding 1,0 
 z  1  0, bottom of fraction  0 

26. z 1
ii  Find the locus of z if w 
and w is purely real
z 1
 x  1  iy  x  1  iy OR If w is purely real then arg w  0 or 
w

 x  1  iy  x  1  iy
 z  1   0 or 
x 2  1  i x  1 y  i x  1 y  y 2
i.e. arg


2
 z 1
 x  1  y 2
If w is purely real then Imw  0
i.e.   x  1 y   x  1 y  0
 xy  y  xy  y  0
 2y  0
y0
 locus is y  0, excluding 1,0 
 z  1  0, bottom of fraction  0 
arg z  1  arg z  1  0 or 
y
x

27. z 1
ii  Find the locus of z if w 
and w is purely real
z 1
 x  1  iy  x  1  iy OR If w is purely real then arg w  0 or 
w

 x  1  iy  x  1  iy
 z  1   0 or 
x 2  1  i x  1 y  i x  1 y  y 2
i.e. arg


2
 z 1
 x  1  y 2
If w is purely real then Imw  0
i.e.   x  1 y   x  1 y  0
 xy  y  xy  y  0
 2y  0
y0
 locus is y  0, excluding 1,0 
 z  1  0, bottom of fraction  0 
arg z  1  arg z  1  0 or 
y
-1
1
x

28. z 1
ii  Find the locus of z if w 
and w is purely real
z 1
 x  1  iy  x  1  iy OR If w is purely real then arg w  0 or 
w

 x  1  iy  x  1  iy
 z  1   0 or 
x 2  1  i x  1 y  i x  1 y  y 2
i.e. arg


2
 z 1
 x  1  y 2
If w is purely real then Imw  0
i.e.   x  1 y   x  1 y  0
 xy  y  xy  y  0
 2y  0
y0
 locus is y  0, excluding 1,0 
 z  1  0, bottom of fraction  0 
arg z  1  arg z  1  0 or 
y
-1
1
x

29. z 1
ii  Find the locus of z if w 
and w is purely real
z 1
 x  1  iy  x  1  iy OR If w is purely real then arg w  0 or 
w

 x  1  iy  x  1  iy
 z  1   0 or 
x 2  1  i x  1 y  i x  1 y  y 2
i.e. arg


2
 z 1
 x  1  y 2
If w is purely real then Imw  0
i.e.   x  1 y   x  1 y  0
 xy  y  xy  y  0
 2y  0
y0
 locus is y  0, excluding 1,0 
 z  1  0, bottom of fraction  0 
arg z  1  arg z  1  0 or 
y
-1
1
x

30. z 1
ii  Find the locus of z if w 
and w is purely real
z 1
 x  1  iy  x  1  iy OR If w is purely real then arg w  0 or 
w

 x  1  iy  x  1  iy
 z  1   0 or 
x 2  1  i x  1 y  i x  1 y  y 2
i.e. arg


2
 z 1
 x  1  y 2
If w is purely real then Imw  0
i.e.   x  1 y   x  1 y  0
 xy  y  xy  y  0
 2y  0
y0
 locus is y  0, excluding 1,0 
 z  1  0, bottom of fraction  0 
arg z  1  arg z  1  0 or 
y
-1
1
x
locus is y  0, excluding  1,0 

31. z 1
ii  Find the locus of z if w 
and w is purely real
z 1
 x  1  iy  x  1  iy OR If w is purely real then arg w  0 or 
w

 x  1  iy  x  1  iy
 z  1   0 or 
x 2  1  i x  1 y  i x  1 y  y 2
i.e. arg


2
 z 1
 x  1  y 2
If w is purely real then Imw  0
i.e.   x  1 y   x  1 y  0
 xy  y  xy  y  0
 2y  0
y0
 locus is y  0, excluding 1,0 
 z  1  0, bottom of fraction  0 
arg z  1  arg z  1  0 or 
y
-1
1
x
locus is y  0, excluding  1,0 
Note : locus is y  0, excluding 1,0  only
i.e. answer the original question

32.  z 
iii  Find the locus of z if arg

 z  4 6

33.  z 
iii  Find the locus of z if arg

 z  4 6
 z 
arg

 z  4 6

34.  z 
iii  Find the locus of z if arg

 z  4 6
 z 
arg

 z  4 6

35.  z 
iii  Find the locus of z if arg

 z  4 6
 z 
arg

 z  4 6

arg z  arg z  4  
6
y
x

36.  z 
iii  Find the locus of z if arg

 z  4 6
 z 
arg

 z  4 6

arg z  arg z  4  
6
y
4x

6

37.  z 
iii  Find the locus of z if arg

 z  4 6
 z 
arg

 z  4 6

arg z  arg z  4  
6
y
4x

6
NOTE: arg z  arg z-4 
 below axis

38.  z 
iii  Find the locus of z if arg

 z  4 6
 z 
arg

 z  4 6

arg z  arg z  4  
6
y
2
4x

6
NOTE: arg z  arg z-4 
 below axis

39.  z 
iii  Find the locus of z if arg

 z  4 6
 z 
arg

 z  4 6

arg z  arg z  4  
6
y
2
r 4x
(2,y)

6
NOTE: arg z  arg z-4 
 below axis

40.  z 
iii  Find the locus of z if arg

 z  4 6
 z 
arg

 z  4 6

arg z  arg z  4  
6
y
2
r 4x
(2,y)

6
NOTE: arg z  arg z-4 
 below axis
30

41.  z 
iii  Find the locus of z if arg

 z  4 6
y
 z 
 tan 60
arg

2
 z  4 6
arg z  arg z  4  
y

6
2
r 4x
(2,y)

6
NOTE: arg z  arg z-4 
 below axis
30

42.  z 
iii  Find the locus of z if arg

 z  4 6
y
 z 
 tan 60
arg

2
 z  4 6

y  2 tan 60
arg z  arg z  4  
6
2 3
y
2
r 4x
(2,y)

6
NOTE: arg z  arg z-4 
 below axis
30

43.  z 
iii  Find the locus of z if arg

 z  4 6
y
 z 
 tan 60
arg

2
 z  4 6

y  2 tan 60
arg z  arg z  4  
6
2 3
y
 centre is 2,2 3 
2
r 4x
(2,y)

6
NOTE: arg z  arg z-4 
 below axis
30

44.  z 
iii  Find the locus of z if arg

 z  4 6
y
 z 
 tan 60
arg

2
 z  4 6

y  2 tan 60
arg z  arg z  4  
6
2 3
y
 centre is 2,2 3 
2
r 4x
(2,y)

6
NOTE: arg z  arg z-4 
 below axis
30
r 2  2 2  2 3 
2

45.  z 
iii  Find the locus of z if arg

 z  4 6
y
 z 
 tan 60
arg

2
 z  4 6

y  2 tan 60
arg z  arg z  4  
6
2 3
y
 centre is 2,2 3 
2
r 4x
(2,y)

6
NOTE: arg z  arg z-4 
 below axis
30
r 2  2 2  2 3 
2
r 2  16
r4

46.  z 
iii  Find the locus of z if arg

 z  4 6
y
 z 
 tan 60
arg

2
 z  4 6

y  2 tan 60
arg z  arg z  4  
6
2 3
y
r 2  2 2  2 3 
2
r 2  16
r4
 centre is 2,2 3 
 locus is the major arc of the circle
2
r 4x
(2,y)

6
NOTE: arg z  arg z-4 
 below axis
 x  2   y  2 3   16 formed by the
chord joining 0,0  and 4,0  but not
2
2
30 including these points.

47.  z 
iii  Find the locus of z if arg

 z  4 6
y
 z 
 tan 60
arg

2
 z  4 6

y  2 tan 60
arg z  arg z  4  
6
2 3
y
r 2  2 2  2 3 
2
r 2  16
r4
 centre is 2,2 3 
 locus is the major arc of the circle
2
r 4x
(2,y)

6
NOTE: arg z  arg z-4 
 below axis
 x  2   y  2 3   16 formed by the
chord joining 0,0  and 4,0  but not
2
2
30 including these points.
Patel: Exercise 4N; 5, 6
Cambridge: Exercise 1F; 10 to 20
HSC Geometrical Complex Numbers Questions